2013 June 5
PART 1B: Poincaré's words are in boldface.
According to Lorentz and Fitzgerald, all the bodies borne along in the motion of the earth undergo a deformation. This deformation is, in reality, very slight, since all dimensions parallel to the movement of the earth diminish by a hundred millionth, while the dimensions perpendicular to this movement are unchanged.
But it matters little that it is slight ; that it exists suffices for the conclusion I am about to draw. Moreover, I have said it was "slight", but in reality I know nothing about it; I have myself been victim of the tenacious illusion which makes us believe that we can conceive an absolute space; I have thought of the motion of the earth in its elliptic orbit around the sun, and I have allowed thirty kilometers as its velocity. But its real velocity ( I mean, this time, not its absolute velocity, which is meaningless, but its velocity with relation to the ether) I do not know, and have no means of knowing it: It may be 10 or 100 times greater, and then the deformation will be 100 or 10,000 times more.
Can we show this deformation ? Evidently not. Here is a cube with its edges 1 meter in length ; in consequence of the earth's displacement it is deformed, the edge which is parallel to the motion becoming smaller, the others remaining unchanged. If I wish to assure myself of this by the aid of a meter measure, I shall first measure one of the edges perpendicular to the motion and shall find that my standard meter fits this edge exactly ; and in fact neither of these two lengths is changed, since both are perpendicular to the motion. Then I wish to measure the other edge, that parallel to the motion; to do this I change the position of my meter and turn it so as to apply it to the edge. But since the meter has changed orientation and become parallel to the motion, it has undergone, in its turn, the deformation, so that though the edge be not exactly a meter long, it will fit exactly and I shall find out nothing.
You ask then of what use is the hypothesis of Lorentz and Fitzgerald if no experiment can make its verification possible? It is my exposition that has been incomplete. I have spoken only of measurements that can be made with a meter; but we can also measure a length by the time it takes light to traverse it, on condition that we suppose the velocity of light constant and independent of direction.
Lorentz could have accounted for the facts by supposing the velocity of light greater in the direction of the earth's motion than in the perpendicular direction. He preferred to suppose that the velocity is the same in these different directions, but that the bodies are smaller in the one than in the other.
If the wave surfaces of light had undergone the same deformations as the material bodies we should never have perceived the Lorentz-Fitzgerald deformation.
In either case it is not a question of absolute magnitude but of the measure of this magnitude by means of some instrument which may be a meter or the path traversed by light. It is only the relation of the magnitude to the instrument that we measure ; and if this relation is altered, we have no way of knowing whether it is the magnitude or the instrument which has changed. But what I wish to bring out is that in this deformation the world has not remained similar to itself ; squares have become rectangles, circles ellipses, spheres ellipsoids.
And yet we have no way of knowing whether this deformation is a real one. Evidently we might go much further. In place of the Lorentz-Fitzgerald deformation whose laws are particularly simple, we might imagine any deformation whatsoever. Bodies might be deformed according to any laws however complicated and we should never notice it provided all bodies without exception were deformed according to the same laws. In saying "all bodies without exception" I include of course our own body and the light rays emanating from different objects.
If we look at the world in one of those mirrors of complicated shape which deform objects in a bizarre way, the mutual relations of the different parts of this world would not be altered; if in fact two real objects touch, their images likewise seem to touch. Of course when we look in such a mirror we see indeed the deformation, but this is because the real world continues to exist alongside of its deformed image.
Then, too, even if this real world were hidden from us, there is one thing that could not be hidden, and that is ourself ; we could not cease to see, or at least to feel, our body and our limbs which have not been deformed and which continue to serve us as instruments of measure. But if we imagine our body itelf deformed in the same way as if seen in the mirror, these instruments of measure in their turn will fail us and the deformation will no longer be ascertainable.
Consider in the same way two worlds as images of one another. Each object P of the world A has a corresponding image P' in the world B. The coordinates of this image P' are determinate functions of those of the object P. Moreover they may be any functions whatsoever; I only suppose them chosen once for all. Between the position of P and that of P' there is a constant relation. What this relation is does not matter ; enough that it be constant.
These two worlds will be indistinguishable one from the other; in other words, the first will be for its inhabitants what the second is for its. This will continue to be the case as long as the two worlds remain strangers to each other.
Suppose we live in a world A and have constructed our science and in particular our geometry. In the meantime the inhabitants of world B will have constructed a science, and as their world is the image of ours their geometry will also be the image of ours or, better, it will be the same as ours. But if some day a window is opened for us upon world B, how we shall pity them! "Poor things," we shall say, "they think they have made a geometry, but what they call so is only a grotesque image of our own; their straights are all twisted, their circles are humped, their spheres have capricious inequalities." And we shall never suspect that they say the same of us, and no one will ever know who is right.
One part of space is not, by itself and in the absolute sense of the word, equal to another part of space ; because if it is so for us, it would not be so for the dwellers in the world B ; and these have just as much right to reject our opinion as we have to condemn theirs. I have elsewhere shown what are the consequences of these facts from the view-point of non- Euclidean geometry and other analogous geometries; to that I do not care to return ; and to-day I shall take a somewhat different point of view.
The failure of repeated Nineteenth-Century attempts to detect the breeze caused by Earth's motion through the æther was of course one of the main unsolved problems in physics around 1900. Fitzgerald's original proposal that the æther exerted some kind of pressure which physically shortened measuring rods "and shrank Fisk's epée to a disc" seemed reasonable at first, but when Lorentz and Larmor noticed that a similar contraction of time would keep Maxwell's equations invariant in a moving reference-frame, it became obvious that something deeper was going on. The so-called Lorentz transformation between frames was the work of many physicists -- including Poincaré, whose formulation eventually became standard. Minkowski, in his famous lecture Raum und Zeit, noted that the Lorentz transformations were a symmetry group of four-dimensional spacetime; it was Poincaré who developed this idea fully.
The "Lorentz group" was defined by Poincaré as the group of rotations in four dimensions (three of space and one of time, where the squared magnitudes of the space and time unit-vectors have opposite sign). The seemingly complicated formulæ of the Lorentz transformation are then merely the standard trigonometric expressions for rotation in a plane familiar to high-school students, awkwardly expressed in terms of the tangent (and with the rotation angle imaginary because of the non-Euclidean geometry of time). Rotations, in mathematics, are isometries -- transformations which leave distances unchanged. (A circle of unit radius still has unit radius after being twisted through an angle.) There are of course many other kinds of isometries -- e.g. translations. (A circle of unit radius still has unit radius after being moved.) It can be shown that all isometries can be generated by reflections, as the two-dimensional example below suggests. In Minkowski's four-dimensional space, the group of all possible isometries is called the Poincaré group, and includes the Lorentz group as a subgroup.
Like all physicists of the early 1900s, Poincaré thought of the Lorentz-Fitgerald contraction as a "shortening" which would would make long objects look stubby. In 1959 Roger Penrose and James Terrell independently worked out how moving objects would actually appear: not at all as Poincaré (in agreement with all other early workers in relativity) described above. The following video simulates this "Terrell rotation"; note particularly the appearance of the cube at 3:33-5:00. An explanation of why Fitgerald-contracted objects look rotated may be found at UPSCALE (the University of Toronto's "Undergraduate Physics Students' Computing And Learning Environment").
THE RELATIVITY OF SPACE, by Henri Poincaré